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Constructive Wall-Crossing

PiljinYi (KIAS)

Yongpyung 2012

1102.1729 with Sungjay Lee 1107. 0723 with Heeyeon Kim, Jaemo Park, ZhaolongWang wall-crossing is breaking

supersymmetry is a square-root of translational

supercharges are square-root of Hamiltonian wall-crossing is supersymmetry breaking

supersymmetry breaking = positive vacuum wall-crossing is supersymmetry breaking

prototype : (anomalous) N=1 SQED

integrate out auxiliary D wall-crossing is supersymmetry breaking

prototype : (anomalous) N=1 SQED wall-crossing is supersymmetry breaking

with positive charges

- sidesusy preserved susy broken + side

: a wall that separates susy region from susy broken region wall-crossing is supersymmetry breaking

existence of such walls is a generic feature of systems with 4 or less supercharges

susy broken susy preserved

: a wall that separates susy region from susy broken region wall-crossing occurs in

• D=4 systems:

N=1 supersymmetric field theories with anomalous U(1)

type IIB on Calabi-Yau with 3-form fluxes

type IIA string theory on Calabi-Yau + half-wrapped D6

Heterotic/type I on Calabi-Yau with gauge bundles …….

in other words : entire String Theory Landscape of phenomenological relevance for our wall-crossing of BPS

• D=1 systems (namely, -like states):

supersymmetric kinks in D=2 N=(2,2) theories

supersymmetric (BPS) states in D=4 N=2 Seiberg-Witten theory

supersymmetric (BPS) black holes in D=4 N=2 supergravity

completely wrapped D-branes in type II string theories on Calabi-Yau canonical example : D=4 N=2 Seiberg-Witten SU(2)  U(1) 1996 Bilal + Ferarri self-consistent minimal set of BPS states to SU(2) Seiberg-Witten via monodromy

 a conjecture for the complete BPS spectrum why does the wall-crossing happen and is the conjecture true ? in other D=4 N=2 Seiberg-Witten theories and supergravities ?

? 1998 Lee + P.Y. N=4 SU(n) ¼ BPS states via multi-center classical dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y. N=2 SU(n) BPS states via multi-center monopole dynamics

2001 Denef: N=2 supergravity via classical multi-center black holes attractor solutions a generic BPS state is a loose bound state of many charge centers outside particles are loose bound states of inside particles

wall-crossing = dissociation of supersymmetric bound states wall-crossing = disappearance of supersymmetry particle states  bound states of many charge centers become infinitely large therefore, if viewed from the opposite side wall-crossing backward  appearance of supersymmetric multi-center bound states how to count & classify such supersymmetric bound states 2000 Stern + P.Y. general wall-crossing formula for simple total magnetic charges; weak coupling regime

2002 Denef: quiver dynamics representation of N=2 supergravity BH’s

…………

2008 de Boer + El Showk + Messamah + van den Bleeken 2- & 3- particle bound state conjecture

2010/2011 Manschot + Pioline + Sen: general n-particle conjecture

2011 S. Lee + P.Y. / H. Kim + J. Park + Z. Wang + P.Y. reformulation and full solution via generic dyon/BH dynamics constructive and first-principle derivation of complete wall-crossing formula from Seiberg-Witten theory; origin of rational invariants explained from dynamics main result (i) : primary index formula for generic n-body problems

Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 main result (ii) : bound state counting with Bose/Fermi statistics incorporated

Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 from which we find, for example, and which results in BPS states for extended superalgebra in D=4 field theory

under the angular momentum/Lorentz boost,

transform as a vector and a Dirac spinor, respectively BPS multiplet in rest frame BPS multiplet in rest frame BPS multiplet in rest frame BPS multiplet in rest frame

[ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] 2nd helicity trace as the BPS state counter

[ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] 2nd helicity trace as the BPS state counter

[ a spin ½ + two spin 0 ] x [ angular momentum multiplet ] non-BPS multiplet in rest frame

[ a spin 1 + two spin ½ + five spin 0 ] x [ angular momentum multiplet ] more generally, we wish to compute the protected spin character D=4 N=2 superymmetric field theory

the smallest multiplet with massless vector: Coulomb phase

all off-diagonal fields are massive and can be integrated out D=4 N=2 superymmetric field theory

for SU(r+1) D=4 N=2 superymmetric field theory

related by extended supersymmetry D=4 N=2 superymmetric field theory SU(2)  U(1) case

where off-diagonal particles of mass is already integrated out SU(2)  U(1) case SU(2)  U(1) case BPS particles: electrically / magnetically charged particles with partially preserved susy D=4 N=2 central charge with electric charges and magnetic charges D=4 N=2 central charge with electric charges and magnetic charges let us construct general BPS bound states as a preliminary exercise, consider one dynamical dyon at a time a probe charge to a system of background “core” dyons

background dyons:

one probe dyon: one dynamical charge in the background dyons a probe charge to a system of background “core” dyons a probe charge to a system of background “core” dyons

take one last step: what happens when the imaginary part is very small ? wall-crossing in the core-probe approximation mini wall-crossing problem similar to Schroedinger problems

+side - side wall-crossing in the core-probe approximation

one must elevate this to a N=4 supersymmetric N=4 SUSY QM with 3 & 4 ?

the easiest way to understand this set of multiplets

is as a dimensional reduction of on-shell N=1 D=4 vector multiplet (in the Wess-Zumino gauge) N=4 SUSY QM with 3 bosons & 4 fermions ?

the easiest way to understand this set of multiplets

is as a dimensional reduction of on-shell N=1 D=4 vector multiplet (in the Wess-Zumino gauge)

Smilga; Ivanov; Papadopoulos; …… circa1988-1991 N=4 SUSY QM with 3 bosons & 4 fermions full wall-crossing problem : counting BPS bound states of an arbitrary collection of dyons

+side - side N=4 SUSY QM with 3n bosons & 4n fermions ?

position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ?

position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ?

position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ?

position of A-th dyon N=4 SUSY QM with 3n bosons & 4n fermions ?

is N=4 supersymmetric if and only if N=4 SUSY QM with 3n bosons & 4n fermions ?

is N=4 supersymmetric with N=4 quantum mechanics for aribtrary collections of dyons

Kim+Park+P.Y.+Wang 2011 counting BPS Bound States with N=4 SUSY QM counting BPS bound states with N=4 SUSY QM

supersymmetry  zero energy  wavefunction supported on submanifold ? counting BPS bound states with N=4 SUSY QM

supersymmetric lowest landau level problem on submanifold ? wall-crossing problem  bound state counting  Witten index

for this, only one of the four is needed

 we are allowed to deform the dynamics as long as this preserves one supersymmetry and does not alter the asymptotics dynamics wall-crossing problem  bound state counting  Witten index decoupling of radial directions, again ?

heavy light N=4 Index reduces to a N=1 Index on

3n bosons + 4n fermions  2n bosons + 2n fermions index

trivial for a complete intersection in flat ambient space index Kim+Park+P.Y.+Wang 2011

conjectured and computed for many examples Manschot, Pioline, Sen 2010/2011 symmetry and indices of N=4 quantum mechanics basic index of individual BPS particles symmetry and indices of deformed & reduced quantum mechanics symmetry and indices of deformed & reduced quantum mechanics symmetry and indices of deformed & reduced quantum mechanics connecting to the desired index problem Kim+Park+P.Y.+Wang 2011

deformation connecting to the desired index problem Kim+Park+P.Y.+Wang 2011

deformation connecting to the desired index problem Kim+Park+P.Y.+Wang 2011

deformation so far, construction was for a set of distinguishable particles but Bose/Fermi statistics from identical constituent particles is essential, for example, to solve SU(2) U(1) problem. incorporating Bose/Fermi statistics

free bulk + sum over fixed submanifolds incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics

 solution by iteration if there is a universal answer for incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics incorporating Bose/Fermi statistics

how ? why ?

P.Y. 1997 / Green+Gutperle 1997 incorporating Bose/Fermi statistics final answer with Bose/Fermi statistics incorporated

Kim+Park+P.Y.+Wang 2011 Manschot, Pioline, Sen 2010/2011 for example with a pair of states only on the + side which results in, as promised, proved

“Coulomb phase” wall-crossing conjecture

a la Manschot, Pioline, Sen 2010/2011

from a first principle computation this does not represent the simplest solution to su(2) problem, but it is the most direct derivation of the spectrum from field theory;

beyond SU(2), clever mathematical solution are less effective;

this approach offers an unique route to computerize the search summary

• universal moduli dynamics for BPS dyons/BHs in D=4 N=2 theories

• reduction to classical moduli space with reduced supersymmetry

• index theorems for all non-threshold bound states

• geometric and universal realization of Manschot et.al.’s rational invariants and …

• a more intelligent way of evaluating the master formula ?

• mathematical equivalence to Konsevitch-Soibelmann etc ?

• what’s hidden in the “Higgs” phase ? : closer relationship to D=4 N=1field theory wall-crossing

• D = 4 N=1 systems : global view on D-term susy breaking pattern in the Landscape ? and …

• a more intelligent way of evaluating the master formula ?

• mathematical equivalence to Konsevitch-Soibelmann etc ? Ashoke Sen 1112.2515 • what’s hidden in the “Higgs” phase ? : closer relationship to D=4 N=1field theory wall-crossing

• D = 4 N=1 systems : global view on D-term susy breaking pattern in the Landscape ?